For which closed smooth manifolds does the action of the diffeomorphism group on the set of almost complex structures have exactly one orbit?
For example it is true for $S^2$.
You were lucky to find the only possible example. If you take any manifold of dimension $\ge 4$ you can pick an almost complex structure that is integrable in some closed ball and make it non-integrable outside of it. Since complex $n$-balls have more than one holomorphic structure, we are done. And all surfaces apart from $S^2$ have moduli of complex structures.
Dmitri's answer is fine, but there is a different argument that is purely local that is worth bearing in mind as well:
On a $2n$-manifold $M$, the set of almost complex structures on $M$ are the sections of a smooth bundle $\mathscr{J}(M)\to M$ whose fibers are diffeomorphic to $\mathrm{GL}(2n,\mathbb{R})/\mathrm{GL}(n,\mathbb{C})$, a space of real dimension $4n^2 - 2n^2 = 2n^2$.
Thus, almost complex structures in dimension $2n$ depend locally on $2n^2$ functions of $2n$ variables, while diffeomorphisms of $M$ depend locally on $2n$ functions of $2n$ variables. Since $2n^2>2n$ when $n>1$, it follows that, when $n>1$, almost complex structures have local invariants, i.e., the diffeomorphism group cannot act transitively on the space of $k$-jets of almost complex structures for $k$ sufficiently large. Hence, not all almost complex structures can be equivalent under diffeomorphism when $n>1$, even locally.